Statics, Dynamics, and Rheological Properties of
Micellar Solutions by Computer Simulation
Thèse en cotutelle avec l’
U
niversité
L
ibre de
B
ruxelles
THÈSE
présentée à
l’Université Paul VerlaineMetz
par
ChienCheng HUANG
pour l’obtention du grade de Docteur
Spécialité : Physique
Soutenue le 25 septembre 2007
devant la commission d’examen
M. Wim
Briels
Professeur
,
Twente University, PaysBas
Rapporteur
M. Roland G. Winkler
Professeur, Forschungszentrum Jülich, Allemagne Rapporteur
M. Michel Mareschal
Professeur
,
Université Libre de Bruxelles
Examinateur
M. Joachim Wittmer
D
R CNRS
,
Institut Charles Sadron, Strasbourg
Examinateur
Mme
Hong Xu
Professeur
,
Université Paul VerlaineMetz
Directeur de Thèse
M. JeanPaul Ryckaert
Professeur
,
Université Libre de Bruxelles
Directeur de Thèse
Remerciements
Dans le cadre d’une thèse en cotutelle entre le Laboratoire de Physique des Milieux
Denses de l’Université Paul VerlaineMetz et le Laboratoire de Physique des
Polymères de l’Université de Libre de Bruxelles, j’ai été amené à travailler au sein de
deux laboratoires, ce qui fut une expérience très enrichissante tant d’un point de vue
scientifique qu’humain.
J’exprime, tout d’abord, ma reconnaissance envers le Prof. Hong Xu pour sa
pédagogie, sa grande disponibilité et son soutien ainsi que pour la formation qu’elle
m’a dispensé. Je tiens à remercier ensuite le Prof. JeanPaul Ryckaert pour toutes les
discussions productives que nous avons eues ensemble et le partage de son savoir
dans le domaine de la physique. Je les remercie pour leur soutien jour après jour et
pour leur qualité humaine.
Merci à vous deux de m’avoir donné un si bel exemple de rigueur et d’honnêteté
scientifique.
Je remercie également le Dr. Joachim Wittme et le Prof. Marc Baus pour les
nombreuses idées sur les voies à explorer ainsi que pour les discussions très
enrichissantes.
Je remercie grandement Monsieur Georges Destrée pour les aides sur les techniques
de simulations.
Je souhaite aussi remercier Monsieur François Crevel pour les nombreuses
discussions.
Je voudrais également remercier l’ensemble des membres du jury de thèse, et en
particulier le Prof. Roland G. Winkler et le Prof. Wim Briels pour avoir accepté la
charge de rapporteurs de cette thèse. Merci au Prof. Michel Mareschal d’avoir bien
voulu présider le jury et Dr. Joachim Wittme pour l’intérêt qu’il a porté à ce travail.
Enfin, je tiens également à remercier ma famille et mes proches pour leur confiance et
leur soutien constant.
Abstract
Statics,dynamics,rheology and scission recombination kinetics of self as
sembling linear micelles are investigated at equlibriumstate and under shear
ﬂowby computer simulations using a newly proposed mesoscopic model.We
model the micelles as linear sequences of Brownian beads whose spacetime
evolution is governed by Langevin dynamics.A Monte Carlo algorithm
controls the opening of a bond or the chainend fusion.A kinetic param
eter ω modelling the eﬀect of a potential barrier along a kinetic path,is
introduced in our model.For equilibrium state we focus on the analysis
of short and long time behaviors of the scission and recombination mech
anisms.Our results show that at time scales larger than the life time of
the average chain length,the kinetics is in agreement with the meanﬁeld
kinetics model of Cates.By studying macroscopic relaxation phenomena
such as the average micelle length evolution after a Tjump,the monomer
diﬀusion,and the zero shear relaxation function,we conﬁrm that the ef
fective kinetic constants found are indeed the relevant parameters when
macroscopic relaxation is coupled to the kinetics of micelles.For the non
equilibrium situation,we study the coupled eﬀects of the shear ﬂow and the
scissionrecombination kinetics,on the structural and rheological properties
of this micellar system.Our study is performed in semidilute and dynam
ically unentangled regime conditions.The explored parameter ω range is
chosen in order for the life time of the average size chain to remain shorter
than its intrinsic (Rouse) longest relaxation time.Central to our analysis
is the concept of dynamical unit of size Λ,the chain fragment for which the
life time τ
Λ
and the Rouse time are equal.Shear thinning,chain gyration
tensor anisotropy,chain orientation and bond stretching are found to de
pend upon the reduced shear rate β
Λ
= ˙γτ
Λ
while the average micelle size
is found to decrease with increasing shear rate,independently of the height
of the barrier of the scissionrecombination process.
Contents
Introduction 1
1 Theoretical Framework 7
1.1 Statistical mechanics derivation of the distribution of chain lengths...7
1.2 A kinetic model for scissions and recombinations.............13
1.2.1 Macroscopic thermodynamic relaxation..............16
1.3 Linear viscoelasticity of unentangled micelles...............17
1.3.1 The Rouse model...........................18
1.3.2 The Rouse model implications on the viscous response of a monodis
perse polymer solution........................19
1.3.3 The theory of Faivre and Gardissat and the viscoelasticity of
micelles................................20
2 The mesoscopic model of wormlike micelles 23
2.1 The potential [19]...............................24
2.2 The Langevin Dynamics...........................25
2.2.1 Nonequilibrium LD technique....................28
2.3 MonteCarlo procedure............................29
3 Equilibrium Properties 35
3.1 Static properties...............................36
3.1.1 List of simulation experiments and chain length distribution...36
3.1.2 Chain length conformational analysis................38
3.1.3 Pair correlation function of chain ends and the distribution of
bond length..............................40
3.2 Dynamic properties..............................42
i
3.2.1 Kinetics Analysis...........................43
3.2.2 Analysis of the Monomer Diﬀusion.................65
3.2.3 Zeroshear stress time autocorrelation functions..........71
3.2.4 Macroscopic relaxation behavior..................73
3.3 General comments..............................74
4 Nonequilibrium properties 77
4.1 Collective rheological behavior.......................78
4.1.1 Orientation of the chains......................78
4.1.2 Viscosity...............................82
4.2 Chain length distribution and chain size dependent properties......85
4.2.1 Eﬀect of shear ﬂow on distribution of chain lengths........85
4.2.2 Saturation eﬀect on orientational properties............87
4.3 Scissionrecombination kinetics under ﬂow.................92
5 Conclusions 99
Bibliography 105
ii
Introduction
In the ﬁeld of selfassembling structures,supramolecular polymers are attracting
nowadays much attention [1,56].The terminology of supramolecular polymer applies
to any polymerlike ﬂexible and cylindrical superstructure obtained by the reversible
linear aggregation of one or more type of molecules in solution or in melt.These
supramolecular polymers are typical soft matter systems and the chain length dis
tribution which determines their properties,is very sensitive to external conditions
(temperature,concentration,external ﬁelds,salt contents,etc...).Wormlike micelles
are one of the most common type of supramolecular polymer [2].Selfassembled stacks
of discotic molecules [20] and chains of bifunctional molecules [21] are other examples.
All these examples diﬀer by the nature of the intermolecular forces involved in the self
assembling of the basic units,but they lead to a similar physical situation bearing much
analogy with a traditional system of polydisperse ﬂexible polymers when their length
becomes suﬃciently large with respect to their persistence length.The speciﬁcity and
originality of these supramolecular polymers comes from the fact that these chains are
continuously subject to scissions at random places along their contour and subject to
end to end recombinations,leading to a dynamical equilibrium between diﬀerent chain
lengths species.
It is well known and schematically illustrated in ﬁgure 1 that some surfactant
molecules in solution can selfassemble and form wormlike micelles [2].As their mass
distribution is in thermal equilibrium,they are therefore sometimes termed “equilib
rium polymers” (EP) [1].Similar system of equilibrium polymer are formed by liquid
sulfur [33,34,35],selenium [36] and some protein ﬁlaments [37]
The micellar solutions exhibit fascinating rheological behavior which has been re
cently reviewed and discussed theoretically [56].Quite commonly,under shear ﬂow
in a Couette cell,an originally isotropic micellar solution undergoes a shearbanding
1
transition [3],producing a two phases system spatially organized in a concentric man
ner with both phases lying either close to the inner or to the outer cylinder of the
rheometer.As the viscosity of both phases is diﬀerent,a velocity proﬁle with two
slopes is observed through the gap.This nonequilibriumphase separation is the object
of numerous studies [56].The shear thinning and orientational ordering of wormlike
micelles [55] resembles the usual phenomenon observed in polymer solutions but,in
many micellar solutions close to the overlap concentration[4],one can also observe a
shearthickening behavior whose microscopic origin is still a matter of debate nowa
days.Much better understood is the trend of entangled micellar solutions to display a
Maxwell ﬂuid rheological behavior [2].The simple exponential relaxation behavior has
been theoretically explained by the reptationreaction model[56],taking into account
the scissionrecombination mechanism by which local entanglements can be released by
chain scission.The theoretical treatment of the scissionrecombination process within
the rheological theoretical approaches,is usually based on a simple meanﬁeld (MF)
approach in which correlations between successive kinetic events are fully neglected.
The way to take into consideration such correlated transitions has been detailed by
O’Shaughnessy and Yu [12] in order to explain some high frequency features of the rhe
ological behavior of micellar systems.As the latter kinetics model interprets the stress
relaxation as the result of local eﬀects between successive correlated transitions where
chain ends produced by a scission recombine with each other after a small diﬀusive ex
cursion of the chain ends,such a rheological behavior was called “diﬀusioncontrolled”
(DC) by opposition to the standard MF model.
Computer simulations at the mesoscopic scale oﬀers an interesting route towards
the understanding of the generic properties of wormlike micelles solutions on the basis
of a well controlled microscopic (or mesoscopic) model.The ﬁrst accent within the
simulation approach was put on testing the chain length distribution for various algo
rithms producing,on a lattice or in continuum space,temporary linear self assembling
structures.Earlier Monte Carlo simulations using an asymmetric Potts model were
performed to study static properties of equilibrium polymers [38,39,40].Rouault and
Milchev proposed a dynamical Monte Carlo algorithm [41],based on the highly eﬃcient
bond ﬂuctuation model (BFM) [42,43].In great detail,Wittmer and coworkers [13,22]
investigate the static properties of EP in dilute and semidilute solutions using the
same BFM.They conﬁrmed scaling predictions of the mean chain length in dilute and
2
semidilute limits.Rouault [15] veriﬁed the dependence of mean chain length with con
centration using an oﬀlattice Brownian dynamics simulation.Dynamics at equilibrium
and direct simulation of systems in shear ﬂow were also much investigated over the last
ten years.Kroger and coworkers study EP at equilibrium [58] and under shear ﬂow
[17] by Molecular Dynamics using the particular pair potential FENEC model.This
model is a variant of the traditional pair potential of the LJ+FENE type used to model
(permanent) polymers,a pair potential which diverges both as r goes to zero (repulsive
forces) and at a distance r = R
c
where the corresponding attractive force between
neighboring monomers also diverges.The FENEC potential is a truncated form of the
usual form which is set to a constant beyond a certain distance r = r
m
< R
c
value,
creating a ﬁnite potential well for a bounded pair with respect to the unbounded pair
(r > r
m
).In this way,the covalent bond can break or recombine as the pair distance
crosses the r
m
value.Using the same model,Padding and Boek [16] investigated the
recombination kinetics and the stress relaxation of wormlike micellar systems.They
found that at high concentrations,the kinetics is close to a diﬀusioncontrolled mecha
nism.Milchev and coworks [26] study micelle conformations and their size distribution
by an oﬀlattice microscopic model,to study solutions of EP in a lamellar shear ﬂow
while Padding and Boek investigate the inﬂuence of shear ﬂow on the formation of
rings in a EP system using FENEC model [44].All these studies predict a decrease of
the average micelle size as a result of the shear ﬂow.Other simulation studies envisage
that ultimately,various simulations at diﬀerent length and time scale will have to be
combined to ﬁll the huge gap between the atomistic length and time scales where the
precise chemistry is relevant and the mesoscopic scale where rheological properties can
really be probed by simulation.Studies of this type mixing both the atomistic and
mesoscopic approaches to study wormlike micelles rheology are under progress[57].
The aim of our thesis on isotropic wormlike micelles solutions is to analyze,on the
basis of a dynamical simulation at the mesoscopic level,the inﬂuence on the macro
scopic relaxation phenomena of the dynamical coupling between the usual “ﬂexible
polymer” relaxation processes and the “scissionrecombination” kinetics.We will re
strict ourselves to unentangled solutions (working slightly below or above the isotropic
semidilute threshold with a very ﬂexible mesoscopic equilibrium polymer model) to
avoid the prohibitive computer time needed to follow relaxation phenomena governed
by entangled dynamics.The relevance of the MF kinetics model and the microscopic
3
origin of the scissionrecombination rate constants is one important target of our study.
Unentangled supramolecular polymers dynamics was indeed found to be relevant in
selenium rheology.At the occasion of this experimental rheological study,Faivre and
Gardissat [36] proposed an extension of the traditional Rouse theory to include the
scission kinetic eﬀects on the Rouse modes relaxation.They predicted the way the
zero shear rate viscosity decreases as the scission and recombination kinetic constants
increase,introducing the concept of dynamical subunit whose size ﬁxes the relaxation
time which governs the shear modulus relaxation.
The thesis is organized as follows:The theoretical aspects of wormlike micelles
statics (distribution of micelle sizes) and the dynamics (Cates MF kinetic model,Faivre
and Gardissat theory[36])are gathered in chapter I.
In chapter II,we detail the particular mesoscopic Langevin Dynamics model which
is adopted throughout our thesis.Given our aims,this model has the particularly
useful feature that the static properties (in particular the distribution of chain lengths)
and the dynamics of the scissionrecombination can be tuned separately so that we
will often investigate how the relaxation at a unique state point (from a static point of
view) is modiﬁed by tuning the kinetic rate constants.
Chapter III reports the results of a series of simulations at equilibrium,performed
by Langevin Dynamics.Both a dilute and a semidilute state points are treated.We
check our results regarding the distribution of chain sizes with respect to theoretical
prediction and analyze in detail the microscopic origin of the relevant rate constants.
Chapter IV reports a systematic study of a single semidilute state point of micellar
solution under shear ﬂow.Here two kinds of parameters,the shear rate and the scission
recombination rate constants,are systematically varied.We discuss the evolution of
the average micelle size with shear rate and relate it to a diﬀerent evolution of the two
types of rate constants (scission and recombination).We also discuss the nature of
the relevant reduced shear rate and extrapolate our viscosity data to zero shear rate,
allowing us to test the FaivreGardissat theory’s predictions [36].
Our general conclusions are gathered in the last chapter V.
4
Cylindrical micelles
Equilibrium polymer
COARSE−GRAINING
Surfactant
k
k
s
r
B
q
Chain end
ScissionRecombination
E
Figure 1:Some surfactant molecules in solution selfassemble and form long wormlike
micelles which continuously break and recombine.Their mass distribution is,hence,in
thermal equilibriumand they present an important example of the vast class of systems
termed “equilibrium polymers”[1].The free energy E of the (hemispherical) end cap of
these micelles has been estimated [2] to be of order of 10k
B
T.This energy penalty
(together with the monomer density) determines essentially the static properties and
ﬁxes the ratio of the scission and recombination rates,k
s
and k
r
.Additionally,these
rates are inﬂuenced by the barrier height B which has been estimated to be similar to
the end cap energy.Both important energy scales have been sketched schematically as
a function of a generic reaction coordinate q (see chapter 8 of reference [23]).Following
closely the analytical description [2,22,13] these micellar systems are represented in
this study by coarsegrained eﬀective potentials in terms of a standard beadspring
model.The end cap free energy becomes now an energy penalty for scission events,
i.e.,the creation of two unsaturated chain ends.The dynamical barrier is taken into
account by means of an attempt frequency ω = exp(−B/k
B
T).If ω is large,suc
cessive breakage and recombination events for a given chain can be assumed to be
uncorrelated and the recombination of a newly created chain ends will be of standard
meanﬁeld type.On the other hand,the (return) probability that two newly created
chain ends recombine immediately must be particularly important at large ω.These
highly correlated “diﬀusion controlled” [12] recombination events do not contribute to
the eﬀective macroscopic reaction rates which determine the dynamics of the system.
6
Chapter 1
Theoretical Framework
In this chapter we will brieﬂy present the theoretical framework of cylindrical micelles
and some standard polymer theoretical aspects which are pertinent to the ﬂexible
micelles system.In Section 1,we introduce the statistical mechanics derivation of the
distribution of chain lengths and of the corresponding average chain length.Then in
section 2,for the scissionrecombination kinetics at equilibrium,we study the theoretical
formalismof equilibriumpolymers (EP) developed by Cates and coworkers [9,2],based
on a meanﬁeld approximation.And the third section of this chapter will focus on the
linear viscoelasticity of EP.In this section,a framework for linear viscoelacity of dilute
polymer solutions,and intrinsic shear modulus by the Rouse model are presented.
Finally,we brieﬂy review the theory of Faivre and Gardissat [36] where a modiﬁcation
of the standard Rouse theory of linear viscosity of a polydisperse polymer system is
proposed.
1.1 Statistical mechanics derivation of the distribution of
chain lengths
To treat a system of wormlike micelles theoretically,it is convenient to work at the
mesoscopic scale using a model of linear ﬂexible polymers made of L monomers of size
b linked together by a non permanent bonding scheme.Within the system,individual
chain lengths ﬂuctuate by bond scission and by fusion of chain ends of two diﬀerent
chains.Statistical mechanics can be employed to predict the equilibrium distribution
of chain lengths[2].In terms of the equilibriumchain number density c
0
(L),the average
chain length L
0
and the total monomer density φ are given by
7
L
0
=
P
∞
L=0
Lc
0
(L)
P
∞
L=0
c
0
(L)
(1.1)
φ =
∞
X
L=0
Lc
0
(L) =
M
V
(1.2)
where M denotes the total number of monomers in the system and V is the volume.
Conceptually,we consider the Helmholtz free energy F(V,T;{N(L)},N
s
) of a mix
ture of chain molecules of diﬀerent length L in solvent where,in addition to the temper
ature T,the volume V and the number of solvent molecules N
s
,the number of chains
of each speciﬁc length N(L) is ﬁxed.Let F(V,T;M,N
s
) be the Helmholtz free energy
of a similar system where only the total number of solute monomers M is ﬁxed.The
equilibrium chain length distribution c
0
(L) = N(L)/V will result from the set {N(L)}
which satisﬁes the condition
F(V,T;M,N
s
) = min
{N(L)}
"
F(V,T;{N(L)},N
s
) +
∞
X
L=0
LN(L)
#
(1.3)
The parameter is the Lagrange multiplier associated with the constraint that
the individual number of chains N(L) must keep ﬁxed the total number of monomers
M =
P
∞
0
LN(L).Minimization requires that the ﬁrst derivative with respect to any
N(L
′
) variable (L’=1,2,...) is zero,giving
δF(V,T;{N(L)},N
s
)
δN(L
′
)
+L
′
= 0;L
′
= 1,2,...(1.4)
We expect the entropic part of the total free energy F(V,T;{N(L)},N
s
) to be the
sumof translational and chain internal conﬁgurational contributions which both depend
upon the way the M monomers are arranged into a particular chain size distribution.
For the translation part,the polydisperse system entropy is estimated as the ideal
mixture entropy S
id
S
id
(V,T;{N(L)},N
s
) = −k
B
X
L
N(L) ln(CN(L)) +S
solv
(1.5)
where C = b
3
/V is a dimensionless constant independent of L and where S
solv
is
the solvent contribution,independent of the N(L) distribution.The conﬁgurational
entropy of an individual chain with L monomers is written as S
1
(L) = k
B
lnΩ
L
,in
8
1.1 Statistical mechanics derivation of the distribution of chain lengths
terms of Ω
L
,the total number of conﬁgurations of the chain.Adding the conﬁgurational
contributions to S
id
as given by eq.(1.5),the total entropy becomes
S(V,T;{N(L)},N
s
) = −k
B
X
L
N(L) [ln(CN(L)) −lnΩ
L
] +S
solv
(1.6)
We now turn to the energy E(V,T;{N(L)},N
s
) of the same system.If E
1
(V,T;
L) represents the internal energy of a chain of L monomers and E
s
the energy of a
solvent molecule,the energy can be written as
E(V,T;{N(L)},N
s
) =
X
L
N(L)E
1
(V,T;L) +N
s
E
s
(V,T) (1.7)
The key contribution in E
1
is the chain endcap energy E which corresponds to
the chain end energy penalty required to break a chain in two pieces.We will suppose
that E
1
(L) = E + L˜ǫ where ˜ǫ is an irrelevant energy per monomer as M˜ǫ,its total
contribution to the system energy,is independent of the chain length distribution.
The present approximation of the total free energy of the system is thus given by
incorporating in the general expression (1.3) the expressions (1.6) and (1.7),giving
βF(V,T;{N(L)},N
s
) =
X
L
N(L) [lnN(L) +lnC −lnΩ
L
+βE +β˜ǫL] (1.8)
where irrelevant constant solvent terms have been omitted as we only need the ﬁrst
derivative of the free energy with respect to N(L),which now takes the form
δβF
δN(L)
= lnN(L) +lnC −lnΩ
L
+βE +β˜ǫL+1.(1.9)
With this expression,the minimization condition on N(L) becomes
lnN(L) +lnC −lnΩ
L
+βE +1 +
′
L = 0 (1.10)
where
′
= (β +β˜ǫ).We note at this stage that the second derivative of βF(V,T;
{N(L)}) +
′
P
L
LN(L) with respect to N(L) and N(L
′
) variables gives the non neg
ative result
δ
LL
′
N(L)
,indicating that the extremum is indeed a minimum.
Solving for N(L) in eq.(1.10),we get
N(L) = C
′
−1
exp−(
′
L+βE −lnΩ
L
) (1.11)
where C
′
= eC while,according to eq.(1.2)),
′
must be such that
X
L
Lexp−(
′
L+βE −lnΩ
L
) = MC
′
(1.12)
9
The equilibrium N(L) variables are also related to the equilibrium chain length
average L
0
(see eq.(1.1)),so that
X
L
exp−(
′
L+βE −lnΩ
L
) =
MC
′
L
0
(1.13)
To progress,we nowneed to specify the explicit L dependence of Ω
L
.The traditional
single chain theories of polymer physics provide universal expression of Ω
L
in terms of
the polymer size,the environment being simply taken into account through the solvent
quality and the swollen blob size in the semidilute (good solvent) case.
1.1.0.1 The case of mean ﬁeld or ideal chains
The basic meanﬁeld or ideal chain model for a L segments chain gives
Ω
id
L
=
C
1
z
L
(1.14)
where z is the single monomer partition function and C
1
a dimensionless constant.
Adapting eq.(1.11),one has
N(L) =
C
1
C
′
exp−(βE) exp(−”L) (1.15)
where ” =
′
−lnz must,according to eq.(1.12),be such that
X
L
Lexp−(”L) =
1
”
2
=
MC
′
C
1
exp−(βE)
(1.16)
while eq.(1.13) takes the form
X
L
exp−(”L) =
1
”
=
MC
′
L
0
C
1
exp−(βE)
(1.17)
In eqs.(1.16) and (1.17),sums over L from 1 to ∞have been approximated by the
result of their continuous integral counterparts.
Combining eqs.(1.15),(1.16) and (1.17),one gets the ﬁnal expression for the chain
number densities
c
0
(L) =
φ
L
2
0
exp(−
L
L
0
) (1.18)
with the average polymer length given by
L
0
= B
1/2
1
φ
1
2
exp
βE
2
.(1.19)
where B
1
= eb
3
/C
1
is a constant depending upon the monomer size b and the prefactor
in the number of ideal chain conﬁgurations in eq.(1.14).
10
1.1 Statistical mechanics derivation of the distribution of chain lengths
1.1.0.2 The case of dilute chains in good solvent
Polymer solutions are in a dilute regime when chains do not overlap and in semidilute
regime when chains do strongly overlap while the total monomer volume fraction is still
well below its melt value.In the semidilute regime in good solvent condition,chains
remain swollen locally over some correlation length,known as the swollen blob size χ,
but they are ideal over larger distances as a result of the screening of excluded volume
interactions between blobs.
Speciﬁcally,for a given monomer number density φ,the blob size is given by the
condition that the blob volume χ
3
= b
3
L
∗3ν
is equal to the total volume V divided by
the total number of blobs M/L
∗
in the swollen blob.This gives estimates in terms of
the reduced number density φ
′
= b
3
φ,
L
∗
= φ
′
(
1
1−3ν
)
(1.20)
χ = bφ
′
(
ν
1−3ν
)
(1.21)
where ν = 0.588 in present good solvent conditions[8].
In living polymers characterized by a monomer number density φ and some averaged
chain length L
0
,the semidilute conditions correspond to the case L
0
>> L
∗
.We
discuss in this subsection the theory for the dilute case where L
0
<< L
∗
.We will come
back to the semidilute case in the next subsection.
Selfavoiding walks statistics apply to dilute chains in good solvent,and we thus
adopt the number of conﬁgurations[6,7](See especially page 128 of the book of Grosberg
and Khokhlov [7])
Ω
EV
L
= C
1
L
(γ−1)
z
L
(1.22)
for a chain of size L,where γ is the (entropy related) universal exponent equal to 1.165,
z is the single monomer partition function and C
1
a dimensionless constant.
Incorporating expression (1.22) in eq.(1.11),one gets
N(L) =
V
B
1
exp−(βE)L
(γ−1)
exp−(”L) (1.23)
where B was introduced in eq.(1.19) and where ” =
′
−lnz must be ﬁxed by eq.(1.2)
X
L
L
γ
exp−(”L) = B
1
φexp(βE) (1.24)
11
while eq.(1.13) takes here the form
X
L
L
(γ−1)
exp−(”L) =
B
1
φ
L
0
exp(βE) (1.25)
If L is treated as a continuous variable,eqs (1.24) and (1.25) can be rewritten in
terms of the Euler Gamma function satisfying Γ(x) = xΓ(x −1) as
Z
∞
0
L
γ
exp−(”L)dL =
Γ(γ +1)
”
(γ+1)
= B
1
φexp(βE) (1.26)
Z
∞
0
L
(γ−1)
exp−(”L)dL =
Γ(γ)
”
γ
=
B
1
φ
L
0
exp(βE) (1.27)
From eqs (1.26) and (1.27),one gets
” =
γ
L
0
(1.28)
B
1
φexp(βE) =
Γ(γ +1)
γ
(γ+1)
L
(γ+1)
0
(1.29)
These results lead then ﬁnally to the SchulzZimmdistribution of chain lengths,namely
c
0
(L) =
exp(−βE)
B
1
L
(γ−1)
exp(−γ
L
L
0
) (1.30)
and an average polymer length given by
L
0
=
γ
γ
Γ(γ)
1
1+γ
B
1
1+γ
1
φ
1
1+γ
exp
βE
(1 +γ)
.(1.31)
1.1.0.3 The semidilute case
We consider here the semidilute case in good solvent where the average length of living
polymers L
0
is much larger than the blob length L
∗
.The usual picture of a semidilute
polymer solution is an assembly of ideal chains made of blobs of size χ.Using this
approach,Cates and Candau [2] and later J.P.Wittmer et al [13] derived the relevant
equilibrium polymer size distribution.In this subsection,we adapt their derivation to
the theoretical framework presented above.
Let Ω
b
be the number of internal conﬁgurations per blob and z
′
some coordination
number for successive blobs.As there are n
b
= L/L
∗
blobs for a chain of L monomers,
we write the total number of internal conﬁgurations of a chain of size L as
Ω
SD
L
= C
1
L
∗(γ−1)
Ω
L/L
∗
b
z
′
L/L
∗
(1.32)
12
1.2 A kinetic model for scissions and recombinations
where γ is the universal exponent in the excluded volume chain statistics met earlier
for chains in dilute solutions.The important factor L
∗(γ−1)
can be seen as an entropy
correction for chain ends just like E was an energy correction to L˜ǫ.This entropic
term which involves the number of monomers per blob,is needed to take into account
that when a chain breaks,its two ends are subject to a reduced excluded volume
repulsion.The other factors in eq.(1.32) will lead to terms linear in L after taking
the logarithm and thus will be absorbed in the Lagrange multiplier deﬁnition,as seen
earlier in similar cases for ideal and dilute chains.The resulting expression of N(L) in
terms of the Lagrange multiplier (cf.eq.(1.15)) can then be written by analogy as
N(L) =
C
1
C
′
exp−[βE −(γ −1) lnL
∗
] exp(−L) (1.33)
Proceeding as in the ideal case (simply replacing at every step the constant βE
by βE − (γ − 1) lnL
∗
,one recovers in the semidilute case the simple exponential
distribution
c
0
(L) =
φ
L
2
0
exp(−
L
L
0
) (1.34)
with a slightly diﬀerent formula for the average polymer length
L
0
∝ φ
α
exp
βE
2
(1.35)
where α =
1
2
(1 +
γ−1
3ν−1
) is about 0.6.
1.2 A kinetic model for scissions and recombinations
The interest for wormlike micelles dynamics came from the experimental observation
that entangled ﬂexible micelles often display,after an initial strain,a simple exponential
stress relaxation.The mechanism of such relaxation is diﬀerent from that of usual
dead polymer entangled melts.In the latter system,stress relaxation requires that
individual chains leave the strained topological tube created by the entangled temporary
network by a reptation mechanism.A theoretical model,taking into account the extra
relaxation mechanismcaused by scissions and recombinations of micelles,leads [9] to an
exponential decay of the shearing forces with a decay Maxwell time equal to τ ≈
√
τ
b
τ
rep
where τ
rep
is the chain reptation time and τ
b
is the mean life time of a chain of average
size in the system.
13
We will assume in the following that the Cates’s scissionrecombination model gov
erning the population dynamics,originally devised to explain entangled equilibrium
polymer melt rheology,should also apply to the kinetically unentangled regime which
is explored in the present work.
This Cates kinetic model [9] assumes that
• the scission of a chain is a unimolecular process,which occurs with equal proba
bility per unit time and per unit length on all chains.The rate of this reaction is
a constant k
s
for each chemical bond,giving
τ
b
=
1
k
s
L
0
(1.36)
for the lifetime of a chain of mean length L
0
before it breaks into two pieces.
• recombination is a bimolecular process,with a rate k
r
which is identical for all
chain ends,independently of the molecular weight of the two reacting species they
belong to.It is assumed that recombination takes place with a new partner with
respect to its previous dissociation as chain end spatial correlations are neglected
within the present mean ﬁeld theory approach.It results from detailed balance
that the mean life time of a chain end is also equal to τ
b
.
Let c(t,L) be the number of chains per unit volume having a size L at time t.On
the basis of the model,the following kinetic equations can be written [9]
dc(t,L)
dt
= −k
s
Lc(t,L) +2k
s
Z
∞
L
c(t,L
′
)dL
′
+
k
r
2
Z
L
0
c(t,L
′
)c(t,L −L
′
)dL
′
−k
r
c(t,L)
Z
∞
0
c(t,L
′
)dL
′
(1.37)
where the two ﬁrst terms deal with chain scission (respectively disappearance or appear
ance of chains with length L) while the two latter terms deal with chain recombination
(respectively provoking the appearance or disappearance of chains of length L).
It is remarkable that the static solution of this empirical kinetic model leads to an
exponential distribution of chain lengths.Indeed,direct substitution of solution c
0
(L)
in the above equation leads to the detailed balance condition:
φ
k
r
2k
s
= L
2
0
(1.38)
14
1.2 A kinetic model for scissions and recombinations
the ratio of the two kinetic constant being thus restricted by the thermodynamic state.
Detailed balance means that for the equilibrium distribution c
0
(L),the number of
scissions is equal to the number of recombinations per unit time and volume.The total
number of scissions and recombinations per unit volume and per unit time,denoted
respectively as n
s
and n
r
,can be expressed as
n
s
= k
s
φ
L
2
0
Z
∞
0
Lexp(−L/L
0
)dL = k
s
φ (1.39)
n
r
=
k
r
2
φ
2
L
4
0
Z
∞
0
dL
′
Z
∞
0
dL”exp(−
L
′
L
0
) exp(−
L”
L
0
) =
φ
2
k
r
2L
2
0
(1.40)
and it can be easily veriﬁed that detailed balance condition equation 1.38 implies n
s
=
n
r
.
Mean ﬁeld theory assumes that a polymer of length L will break on average after
a time equal to τ
b
= (k
s
L)
−1
according to a Poisson process.This implies that the
distribution of ﬁrst breaking times (equal to the survival times distribution) must be
of the form
Ψ(t) ∝ exp(−
t
τ
b
) (1.41)
for a chain of average size.Detailed balance then requires that the same distribution
represents the distribution of ﬁrst recombination times for a chain end[2].Accordingly,
throughout the rest of this chapter,the symbol τ
b
will represent as well the average
time to break a polymer of average size or the average time between end chain recombi
nations.In the same spirit,we stress that among the diﬀerent estimates of τ
b
proposed
in this work,some are based on analyzing the scission statistics while others are based
on the recombination statistics.
Two additional points may be stressed at this stage:
• The mean ﬁeld model in the present context has been questioned [12] because
in many applications,there are indications that a newly created chain end often
recombines after a short diﬀusive walk with its original partner.In that case,a
possibly large number of breaking events are just not eﬀective and the kinetics
proceeds thus diﬀerently.
• Given the statistical mechanics analysis in the previous subsection,we see that
the equilibrium distribution of chain lengths resulting from the simple empirical
kinetic model is perfectly compatible with the equilibriumdistribution in polymer
15
solutions at the θ point (ideal chains) or for semisolutions (ideal chains of swollen
blobs).The kinetic model is still pertinent,given its simplicity,in the case of
dilute solutions in good solvent as the chain length distribution although non
exponential,is not very far from it.
1.2.1 Macroscopic thermodynamic relaxation
In equation (1.35) the average length L
0
depends on the thermodynamic variables of
the systemsuch as the temperature,the pressure,and the volume fraction of monomers
φ.Under a sudden change in one of the thermodynamic variables (for example,the
temperature),the distribution of the length of the chains in equation (1.37) relaxes
to a new equilibrium exponential distribution characterized by a new average length
L
∞
.The corresponding theory of macroscopic thermodynamic relaxation has been
given by Marques and Cates [27] and tested using a Monte Carlo study by Michev
and Rouault [46].This is in general a complicated,nonlinear decay which can be
monitored experimentally by light scattering which probes the evolution of the average
length.The theory is based on the equations of distribution of length of chain (1.34)
and the kinetic model of Cates [9] i.e.equation (1.37).The characteristic time provides
information about the kinetics of the system.
Marques and Cates [27] show that for any amplitude of the perturbation which
conserves the total volume fraction of monomers (for instance,a temperature modiﬁca
tion having an arbitrary timedependence),the distribution function c(t,L) of equation
(1.37) remains exponential versus L.Indeed,they show that
c(t,L) = φf
2
(t) exp{−Lf(t)} (1.42)
is a nonlinear eigenfunction of equation (1.37) with a eigenvalue f(t) that obeys
df
dt
= k
s
1 −
φk
r
2k
s
f
2
(t)
,(1.43)
where k
r
and k
s
the kinetic constants of the new equilibriumstate,after,e.g.a Tjump.
The time evolution of f(t) for a thermodynamic jump is then the solution of (1.43) with
the initial condition f(t = 0) = 1/L
0
.One obtains [27]:
f(t) = L
−1
∞
tanh
t +t
0
2τ
if L
0
> L
∞
(1.44)
f(t) = L
−1
∞
coth
t +t
0
2τ
if L
0
< L
∞
(1.45)
16
1.3 Linear viscoelasticity of unentangled micelles
where the constant t
0
is given by
t
0
= 2τ tanh
L
∞
L
0
if L
0
> L
∞
(1.46)
t
0
= 2τ coth
L
∞
L
0
if L
0
< L
∞
,(1.47)
and the time
τ =
1
2k
s
L
∞
(1.48)
These equations describe the recovery of c(t,L) and hL(t)i the average chain length,
following a sudden change in temperature which is given by
hL(t)i =
1
f(t)
(1.49)
We see that by measuring the nonequilibrium τ the relaxation time of the mean chain
length,one can get an estimate of the microscopic scission rate constant k
s
.
1.3 Linear viscoelasticity of unentangled micelles
To describe the dynamics of cylindrical micelles in solution,each micelle can be mod
elled as a linear collection of L identical fragments (L being proportional to the length
of the micelle),subject to Brownian motion to represent the dynamical eﬀect of the
bath (the coupling to the rest of the degrees of freedom),subject to connecting forces
between adjacent fragments to enforce the linear structure of the micelle and possibly
subject to pair interactions between non connected fragment pairs.If,as the rela
tive distance between adjacent fragments increases,the connecting tension force is not
bounded to a ﬁnite value to allow an intrinsic possibility of scission (e.g.the FENEC
model of Kroger[17]),the scission and recombination processes must be incorporated
independently in the model.The latter must specify the way through which individ
ual bonds can break (chain scission) or reform (chain end recombinations) by a pair
potential swap for the involved pair of fragments.The micelle dynamics thus needs
to combine all these various ingredients,implying that individual chain entities will
survive for a ﬁnite life time during which ordinary polymer relaxation takes place.
Within a theoretical approach of the dynamical properties of micellar solutions
where the emphasis lies in the search for analytical predictions,it is useful to explore
the validity of the traditional Rouse model to represent the “dead polymer” dynamics
17
part (valid as long as an entity survives),when it can be assumed that excluded volume
and entanglement eﬀects are absent or weak.In the present purely theoretical discus
sion,we thus restrict ourselves to the unentangled dynamics of θ solvent chains.We
brieﬂy review the basic ingredients and essential results of the Rouse model for “dead
chains”,including linear viscoelasticity and we report the extension of this theory to
polymer chains when they are the object of scissions and recombinations,along the
lines proposed by Faivre and Gardissat [36].
1.3.1 The Rouse model
The Rouse model treats all chains as being independent.Consider a “dead chain” of
N elements at temperature T,where each fragment is subject to a friction force pro
portional to its instantaneous velocity with friction coeﬃcient ξ and to a random force
modelled as a 3D white noise stochastic process whose statistical properties are given
by the ﬂuctuationdissipation theorem in terms of T and ξ.The only interactions con
sidered in the Rouse model are harmonic spring forces with spring constant k between
neighboring beads to ensure the linear connectivity.Under this dynamics,the average
squared distance between adjacent monomers deﬁnes a typical mean distance b given
by b
2
= 3k
B
T/k,where k
B
is the Boltzmann constant.The mean squared endtoend
distance is simply (N −1)b
2
.
Chain dynamics considered via the endtoend vector time relaxation function leads
at long times to an exponential decay with the main (Rouse) relaxation time
τ
R
= τ
0
N
2
(1.50)
where τ
0
=
ξb
2
3k
B
Tπ
2
,indicating that the global relaxation time of a chain of size N is
proportional to N
2
.
The way to derive this important result requires the introduction of modes (Rouse
modes),deﬁned as
¯
X
q
(t) =
1
N
N
X
j=1
A
qj
¯
R
j
(t);q = 0,1,2, (N −1) (1.51)
where
¯
R
j
is the coordinate of the j
th
monomer in a chain and
¯
¯
A a square matrix with
elements
A
qj
= cos
qπ
N
(j −
1
2
)
;q = 0,1,2, (N −1);j = 1,2, N (1.52)
18
1.3 Linear viscoelasticity of unentangled micelles
According to this deﬁnition,the mode q = 0 represents the center of mass diﬀusive
motion,while the other (N1) modes q > 0 are associated to the collective motion of
subchains of size N/q.The time autocorrelation of these internal chain modes are
<
¯
X
q
(t)
¯
X
q
(0) >=< X
2
q
> exp
−
t
τ
q
(1.53)
where
τ
q
=
ξb
2
3k
B
Tπ
2
N
q
2
(1.54)
which shows that the Rouse time is the relaxation time associated to the mode q = 1,
while the relaxation time of higher index modes decreases as 1/q
2
.For completeness,
the mode amplitudes are given by
< X
2
q
>=
Nb
2
2π
2
1
q
2
(1.55)
Rouse theory leads to an expression of the time autocorrelation of the chain end
toend vector
¯
S =
¯
R
N
−
¯
R
1
<
¯
S(t).
¯
S(0) >= (N −1)b
2
P′
q
τ
q
exp(−t/τ
q
)
P′
q
τ
q
(1.56)
where the prime in the sums means that the latter only involves the odd q mode
indices.This result shows a typical Rouse theory relaxation function:it is a sum
of terms implying the diﬀerent mode relaxation,which explains that at long times,
only the slowest mode survives and the behavior of the time autocorrelation becomes
exponential with characteristic time τ
R
.
1.3.2 The Rouse model implications on the viscous response of a
monodisperse polymer solution
We now discuss the intrinsic shear modulus G(t) of a monodisperse polymer solution
which reﬂects the viscoelastic response of the system in the linear regime (limit of very
small shear rates).The Newtonian shear viscosity is related to the shear modulus
through
η
0
=
Z
∞
0
G(t)dt (1.57)
Rouse theory for a monodisperse solution of “dead” polymers of size N at monomer
number density φ gives [48]
G(t) =
φ
N
k
B
T
N−1
X
q=1
exp(−2t/τ
q
) =
φ
N
k
B
T
N−1
X
q=1
exp(−2tq
2
/τ
R
) (1.58)
19
This important result indicates that,because of the independence of the dynamics
of the various chains in the system,Rouse theory gives for a collective and intensive
quantity like G(t) an expression which is proportional to the solute concentration while
its viscoelastic behavior is strictly identical to the single chain viscoelastic relaxation
process.Therefore,the shear modulus again appears as a sum of terms expressing the
speciﬁc contributions of the various single chain modes in the relaxation process,with
at long times,a simple exponential time decay with characteristic time τ
R
.By time
integration,according to eq.(1.57),the shear viscosity is then given (for N >> 1) by
η
0
=
π
2
12
φk
B
T
N
τ
R
(1.59)
where τ
R
is given in eq.(1.50).
1.3.3 The theory of Faivre and Gardissat and the viscoelasticity of
micelles
The above subsection is dealing with a monodisperse dilute solution of “dead” chains.
The stress relaxation function of a micellar system is in reality the object of a coupling
between the stress relaxation and the scissionrecombination process if the time scale of
the second process is shorter than the viscoelastic times scales.In our work,we adapt
the theory proposed by Faivre and Gardissat [36],originally developed to interpret
rheological data of liquid selenium.Faivre and Gardissat [36] proposed a modiﬁcation
of the standard Rouse theory of linear viscosity of a polydisperse polymer system [8]
to incorporate the inﬂuence of the scission events.
If we have a polydisperse system of polymers with normalized weight function W
p
,
the relaxation modulus given by the Rouse model should be given by
G(t) =
∞
X
p=1
W
p
G
p
(t) (1.60)
where G
p
(t) concerns the “polymer” part of chain of size p which,within the Rouse
model,reads (see eq.(1.58))
G
p
(t) = G
0
1
p
p
X
q=1
exp
"
−2
t
τ
0
q
p
2
#
,(1.61)
where p is the polymerization degree,G
0
= φk
B
T a material constant,and τ
0
the
local dynamic relaxation time which depends on the solvent viscosity.Notice that for
20
1.3 Linear viscoelasticity of unentangled micelles
physical reasons we have q ≤ p in the sum of Rouse modes for any upper limit p.As
mentioned earlier,the stress relaxation is a superposition of exponentials,each term
corresponds to the contribution to the relaxation of a particular Rouse mode.All terms
have the same amplitude but decay with a characteristic time
τ
p
q
= τ
0
p
2
q
2
(1.62)
which depends on the p/q ratio which is the number of monomers per wave length for
that particular mode.If bond scission occurs independently and uniformly along the
chain with rate k
s
(per unit time and per bond),the lifetime of a chain of p/q monomers
should be
τ
b
=
q
k
s
p
.(1.63)
The Rouse mode q in the original chain of length p should therefore have a survival
probability which decays in time as exp(−k
s
(p/q)t),hence the idea of multiplying the
contribution of each mode to the relaxation of G(t) by this survival probability to take
into account the eﬀect of bond scissions.This leads to the expression
G
′
p
(t) = G
0
1
p
p
X
q=1
exp
"
−k
s
p
q
t −2
t
τ
0
q
p
2
#
.(1.64)
When the life time τ
b
of the chain of average size is shorter than the internal Rouse
relaxation time of the same chain τ
0
L
2
0
,the viscoelastic response is independent of the
mean chain length L
0
but depends upon a new intrinsic time τ
Λ
which corresponds
to the relaxation of a dynamical unit of size Λ deﬁned by the equality of the Rouse
relaxation time τ
0
Λ
2
and the life time,(k
s
Λ)
−1
:
τ
0
Λ
2
= (k
s
Λ)
−1
.(1.65)
It leads to
Λ = (τ
0
k
s
)
−1/3
(1.66)
and
τ
Λ
= τ
1/3
0
k
−2/3
s
(1.67)
so that equation (1.64) can be rewritten as
G
′
p
(t) = G
0
1
p
p
X
q=1
exp
−
t
τ
Λ
"
p
qΛ
+2
qΛ
p
2
#!
.(1.68)
21
Identifying the number of Rouse beads with the number of beads in present work,using
a known distribution of lengths of the chains of our system (1.18),the weight function
for our equilibrium polymers is
W
p
=
1
L
0
exp
−
p
L
0
.(1.69)
So that G(t) for the equilibrium polymers is now given by
G(t) =
∞
X
p=1
W
p
G
′
p
(t).(1.70)
Note that this expression diﬀers slightly from the Faivre and Gardissat ﬁnal expres
sion as apparently,in equation (18) of reference [36],they assume a relaxation time for
a segment of p monomers to be half of the usual Rouse time.(Note also that in their
paper,the symbol τ
0
corresponds to half the time τ
0
used in our work).
22
Chapter 2
The mesoscopic model of
wormlike micelles
The main goal our thesis is to exploit simulation techniques to improve our under
standing of the link between the macroscopic behavior and microscopic aspects of the
structure and the dynamics of selfassembling micellar systems at equilibrium and un
der shear ﬂow.As it is pragmatically impossible to reach large enough scales of length
and time to determine those macroscopic properties using an atomistic level molecular
dynamics simulation approach,we adopt a mesoscopic model and a Langevin Dynamics
approach as done regularly in the studies on micellar systems.As sketched in ﬁgure 1,
micelles can be represented as linear sequences of Brownian beads which,in addition to
their usual Langevin Dynamics spacetime evolution,can either fuse together to form
longer structures or break down into two pieces.The kinetics can be modelled by a
microscopic kinetic MonteCarlo algorithm which generates new bonds between chain
ends adjacent in space or which breaks existing bonds between adjacent monomers.The
free energy E to creat this new end caps for these micelles becomes an energy penalty
for scission events,i.e.,the creation of two unsaturated chain ends.This scission energy
determines the static propeties and inﬂuences the scission and recombination rates,k
s
and k
r
.In this model,the scission energy E of micelles is controlled by a scission en
ergy parameter W which is deﬁned as an additive potential term which inﬂuences the
switching probability between bonded and unbounded potentials.The barrier energy
of recombination B also inﬂuences the kinetic rate constants.The advantage of our
model is that the dynamical barrier height B can be taken into account by means of
the attempt frequency ω associated to the Monte Carlo potential swap.If ω is small,
23
successive breakage and recombination events for a given chain can be assumed to be
uncorrelated and the recombination of newly created chain ends will be of standard
meanﬁeld type.On the other hand,the (return) probability that two newly created
chain ends recombine immediately must be particulary important at large ω.These
highly correlated “diﬀusion controlled” [12] recombination events do not contribute do
the eﬀective macroscopic reaction rates which determine the dynamics of the system.
The mesoscopic model is based on a standard polymer model with repulsive Lennard
Jones interactions between all monomer pairs and an attractive FENE (ﬁnitely extensi
ble nolinear elastic) potential [24] to enforce (linear) connectivity.For the scission and
recombination events,a Monte Carlo procedure is set up to switch back and forth be
tween the bounded potential and the unbounded potential.This new model is thus at
the same time justiﬁed by its link with a standard polymer model and by the advantage
that the scission/recombination attempt frequency is a control parameter governing dy
namics,without aﬀecting the thermodynamic and structural properties of the system.
2.1 The potential [19]
We consider a set of micelles consisting of (noncyclic) linear assemblies of Brownian
particles.Within such a linear assembly,the bond potential U
1
(r) acting between ad
jacent particles is expressed as the sum of a repulsive LennardJones (shifted and trun
cated at its minimum) and an attractive part of the FENE type [24].The pair potential
U
2
(r) governing the interactions between any unbounded pair (both intramicellar and
intermicellar) is a pure repulsive potential corresponding to a simple LennardJones
potential shifted and truncated at its minimum.This choice of eﬀective interactions
between monomers implies good solvent conditions.
Using the Heaviside function Θ(x) = 0 or 1 for x < 0 or x ≥ 0 respectively,explicit
expressions (see ﬁgure 2.1) are
U
2
(r) = 4ǫ
(
σ
r
)
12
−(
σ
r
)
6
+
1
4
Θ(2
1/6
σ −r)) (2.1)
U
1
(r) = U
2
(r) −0.5kR
2
ln
1 −[
r
R
]
2
−U
min
−W (2.2)
In the second expression valid for r < R,k = 30ǫ/σ
2
is the spring constant and
R = 1.5σ is the value at which the FENE potential diverges.U
min
is the minimum
value of the sum of the two ﬁrst terms of the second expression (occurring at r
min
=
24
2.2 The Langevin Dynamics
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
r
10
0
10
20
30
40
50
60
70
80
U
Γ
W
0.96 < r < 1.2
Figure 2.1:Bounded potential U
1
(r) (continuous curve) and unbounded potential U
2
(r)
(dashed line) between a pair of monomers.W is a parameter tuning the energy required
to open the bond.The ﬁgure also shows the Γ region where potential swaps (corre
sponding to bond scissions or bond recombinations) are allowed during the Brownian
dynamics simulation.The unit of length is σ,and for the energy is ǫ.
0.96094σ for the adopted parameters) while W is a key parameter which corresponds
to the typical energy gain (loss) when an unbounded (bounded) pair is the object of a
recombination (scission).
2.2 The Langevin Dynamics
The Langevin equation describes the Brownian motion of a set of interacting particles.
The action of the ﬂuid is split into a slowly evolving viscous force and a rapidly ﬂuc
tuating random force.For a free particle in one dimension the equation is expressed
as
m
i
˙
¯v
i
(t) = −ξ¯v(t)
i
+
¯
F
i
(t) +
¯
R
i
(t) (2.3)
where barv
i
is the velocity of particle i with mass m
i
at time t,
¯
F
i
the systematic force,
ξ the friction coeﬃcient,and
¯
R
i
represents the sum of the forces due to the incessant
collision of the ﬂuid molecules.It is regarded as a stochastic force and is satisfying the
25
condition (ﬂuctuationdissipation theorem) [47]:
< R
iα
(t)R
jβ
(t
′
) >= 2ξk
B
Tδ
ij
δ
αβ
δ(t −t
′
) (2.4)
where the greekletter subscripts αβ refer to the x,y or z components.
2.2.0.1 Brownian Dynamics (BD)
If we assume that the time scales for momentum relaxation and position relaxation
are well separated,then it is possible to consider only time intervals longer than the
momentumrelaxation times.In the diﬀusive regime the momentumof the particles has
relaxed and equation (2.3) can be modiﬁed by setting to zero the momentum variation
(left hand side of equation (2.4)).Then the “position Langevin equation” is given as
d¯r
i
dt
=
¯
F
i
(t)
ξ
+
¯
R
i
(t)
ξ
(2.5)
Earlier simulations within the present work were executed using an algorithm based on
(2.5) [25].The single BD step of particle i subject to a total force
F
i
is simply
¯r
i
(t +Δt) = ¯r
i
(t) +
¯
F
i
ξ
Δt +
¯
Δ
i
(Δt) (2.6)
where the last term given by
¯
Δ
i
(Δt) =
Z
t+Δt
t
¯
R
i
(t
′
)dt
′
/ξ (2.7)
corresponds to a vectorial random Gaussian quantity with ﬁrst and second moments
given by
< Δ
iα
(Δt) > = 0 (2.8)
< Δ
iαβ
(Δt)Δ
jβ
(Δt) > = 2(
k
B
T
ξ
)Δtδ
αβ
δ
ij
(2.9)
for arbitrary particles i and j and where αβ stand for the x,y,z Cartesian components.
At this stage,it is useful to ﬁx units.In the following,we will adopt the size of
the monomer σ as unit of length,the ǫ parameter as energy unit and we will adopt
ξσ
2
/(3πǫ) as time unit.We also introduce the reduced temperature k
B
T/ǫ = T
∗
,so
that in reduced coordinates (written here with symbol*) the random displacement Δ
∗
iα
is
< Δ
∗
iα
> = 0 (2.10)
< Δ
∗
iα
Δ
∗
jβ
> =
2
3π
Δt
∗
δ
αβ
δ
ij
T
∗
(2.11)
26
2.2 The Langevin Dynamics
which means that each particle,if isolated in the solvent,would diﬀuse with a RMSD of
p
2T
∗
/π per unit of time.In the following,all quantities will be expressed in reduced
coordinates without the ∗ symbol.
2.2.0.2 Langevin Dynamics velocity Verlet scheme
Most of the simulation of the present work are performed using Langevin Dynamics
(LD) velocity Verlet algorithmes based on the work of [50].The explict expressions are
¯r
i
(t +Δt) = ¯r
i
(t) +c
1
¯v
i
(t)Δt +c
2
¯
F
i
(t)
m
Δt
2
+δ¯r
G
i
(2.12)
¯v
i(1/2)
= c
0
¯v
i
(t) +
c
0
c
2
c
1
¯
F
i
(t)
m
Δt +δ¯v
G
i
(2.13)
¯v
i
(t +Δt) = ¯v
i(1/2)
+
1
(γΔt)
1 −
c
0
c
1
¯
F
i
(t +Δt)
m
Δt (2.14)
where the ﬁrst equation updates the position,the second equation updates the “half”
step velocities and the third equation completes the velocity move.γ = ξ/m,the
coeﬃcients for the above equations are:
c
0
= e
−γΔt
(2.15)
c
1
= (γΔt)
−1
(1 −c
0
) (2.16)
c
2
= (γΔt)
−1
(1 −c
1
) (2.17)
c
3
= (γΔt)
−1
1
2
−c
1
(2.18)
where e
γΔt
is approximated by its power expansion for γΔt < 1.Each pair of vec
torial component of δr
G
,δv
G
,i.e.δr
G
iα
,δv
G
iα
is sampled from a bivariate Gaussian
distribution[49] deﬁned as
ρ
δr
G
iα
,δv
G
iα
=
1
2πσ
r
σ
v
(1 −c
2
rv
)
1/2
×exp
(
−
1
2(1 −c
2
rv
)
δr
G
iα
σ
r
2
+
δv
G
iα
σ
v
2
−2c
rv
δr
G
iα
σ
r
δv
G
iα
σ
v
!)
(2.19)
with zero mean values,variances given by
σ
2
r
=
D
δr
G
iα
2
E
= Δt
2
k
B
T
m
(γΔt)
−1
2 −(γΔt)
−1
3 −4e
−γΔt
+e
−2γΔt
(2.20)
σ
2
v
=
D
δv
G
iα
2
E
=
k
B
T
m
1 −e
−2γΔt
(2.21)
27
and the correlation coeﬃcient c
rv
determined by
c
rv
=
δr
G
iα
δv
G
iα
= Δt
k
B
T
m
(γΔt)
−1
1 −e
−ξΔt
2
1
σ
r
σ
v
(2.22)
Each pair of cartesian components δr
G
iα
and δv
G
iα
are obtained by the appropriate Gaus
sian distribution according to:
δr
iα
= σ
r
η
1
(2.23)
δv
iα
= σ
v
c
rv
η
1
+η
2
p
1 −c
2
rv
(2.24)
where η
1
and η
2
are two independent random numbers with Gaussian distribution of
zero average and unit variance.
2.2.1 Nonequilibrium LD technique
In this subsection we brieﬂy present the technique adopted for imposing the shear ﬂow
onto the system.We limit the study to a stationary,isovolumic and homogeneous planar
Couette ﬂow characterized by a velocity ﬁeld ¯u(r) =
¯
¯
k
T
r as illustrated in ﬁgure 2.2,
where the velocity gradient
¯
¯
k is constant in space and time.
Figure 2.2:Sketch of planar Couette ﬂow.
For a planar Couette ﬂow in the x direction with a shear along the y axis,the
velocity gradient k is deﬁned as
¯
¯
k =
0 0 0
˙γ 0 0
0 0 0
where ˙γ is shear rate.With the imposition of the solvent velocity ﬁeld,the equation of
motion becomes
m
i
˙
¯v
i
(t) = −mξ
i
(¯v
i
(t) − ¯u(r
i
)) +
¯
F
i
(t) +
¯
R
i
(t) (2.25)
28
2.3 MonteCarlo procedure
A modiﬁcation of the periodic boundary conditions proposed by [51] has been adopted
Figure 2.3:Sketch of shear boundary conditions for planar Couette ﬂow.
for establishing the shear ﬂow.The scheme used is shown in ﬁgure 2.3.In essence,the
inﬁnite periodic system is subjected to a uniform shear in xy plane.The box above
is moving at a speed ˙γL in the positive x direction.The box below moves at a speed
˙γL in the negative x direction.When a particle in the primary box with coordinates
(x,y) crosses a horizontal border one of its images enters from the opposite side at
x
′
= x − ˙γtL
y
with velocity v
′
= v − ˙γL
y
2.3 MonteCarlo procedure
The bonding network is itself the object of random instantaneous changes provided by
a MonteCarlo (MC) algorithm [19] which is built according to the standard Metropolis
scheme [49].
The probability P
m,n
to go from a bounding network m to a diﬀerent one n is
written as
P
m,n
= A
trial
m,n
P
acc
m,n
(2.26)
where A
trial
m,n
is the trial probability to reach a new bounding network n starting from
the old one m,within a single MC step.This trial probability is chosen here to be
symmetric as usually adopted in Metropolis Monte Carlo schemes,and is chosen to be
29
diﬀerent from zero only if both bounding networks n and m diﬀer by the status of a
single bond,say the pair of monomers (ij).To satisfy microreversibilty,the acceptance
probability P
acc
m,n
for the trial (m−→n) must be given by
P
acc
m,n
= Min[1,exp (−
(U({r},n) −U({r},m))
k
B
T
)].(2.27)
In the present case,as a single pair (ij) changes its status,the acceptance probability
takes the explicit form
P
acc
m,n
= Min[1,exp(−
(U
2
(r
ij
) −U
1
(r
ij
))
k
B
T
)] (2.28)
P
acc
m,n
= Min[1,exp(−
(U
1
(r
ij
) −U
2
(r
ij
))
k
B
T
)] (2.29)
for bond scission and bond recombination respectively.
The way to specify the trial matrix A
trial
m,n
starts by deﬁning a range of distances,
called Γ and deﬁned by 0.96 < r < 1.20 within the range r < R.For r ∈ Γ,a
change of bounding is allowed as long as the two restricting rules stated above are
respected.Consider the particular conﬁguration illustrated by ﬁgure 2.4 where M=7
monomers located at the shown positions,are characterized by a connecting scheme
made explicit by representing a bounding potential by a continuous line.The dashed
line between monomers 5 and 6 represents the changing pair with distance r ∈ Γ which
is a bond U
1
(r) in conﬁguration m but is just an ordinary intermolecular pair U
2
(r) in
conﬁguration n.
For further purposes,we have also indicated in ﬁgure 2.4 by a dotted line all pairs
with r ∈ Γ which are potentially able to undergo a change from a non bounded state
to a bounded one in the case where the (56) pair,on which we focus,is non bounded
(state n).Note that the bond (35) is not represented by a dotted line eventhough the
distance is within the Γ range:a bond formation in that case is not allowed as it would
lead to a cyclic conformation.Note also that in state n,monomer 5 could thus form
bonds either with monomer 2 or with monomer 6 while monomer 6 can only form a
bond with monomer 5.
We now state the algorithm and come back later on the special (m−→n) transition
illustrated in ﬁgure 2.4.
During the LDdynamics,with an attempt frequency ω per armand per unit of time,
a change of the chosen arm status (bounded U
1
(r) to unbounded U
2
(r) or unbounded
30
2.3 MonteCarlo procedure
to bounded) is tried.If it is accepted as a “trial move” of the bounding network,it
obviously implies the modiﬁcation of the status of a paired arm belonging to another
monomer situated at a distance r ∈ Γ from the monomer chosen in the ﬁrst place.
The trial move goes as follows:a particular arm is chosen,say arm 1 of monomer
i,and one ﬁrst checks whether this arm is engaged in a bounding pair or not.
• If the selected arm is bounded to another arm (say arm 2 of monomer j) and the
distance between the two monomers lies within the interval r
ij
∈ Γ,an opening
is attempted with a probability 1/(N
i
+1),where the integer N
i
represents the
number of monomers available for bonding with monomer i,besides the monomer
j (N
i
is thus the number of monomers with at least one free arm whose distance
to the monomer carrying the originally selected arm i lies within the interval Γ,
excluding from counting the monomer j and any particular arm leading to a ring
closure).If the trial change consisting in opening the (ij) pair is refused (either
because the distance is not within the Γ range or because the opening attempt
has failed in the case N
i
> 0),the MC step is stopped without bonding network
change (This implies that the LD restarts with the (ij) pair being bonded as
before).
• If the selected arm (again arm 1 of monomer i) is free from bonding,a search is
made to detect all monomers with at least one armfree which lie in the “reactive”
distance range r ∈ Γ from the selected monomer i (Note that if monomer i is a
terminal monomer of a chain,one needs to eliminate from the list if needed,the
other terminal monomer of the same chain in order to avoid cyclic micelles conﬁg
urations).Among the monomers of this “reactive” neighbour list,one monomer
is then selected at random with equal probability to provide an explicit trial
bonding attempt between monomer i and the particular monomer chosen from
the list.Note that if the list is empty,it means that the trial attempt to create a
new bond involving arm 1 of monomer i has failed and no change in the bonding
network will take place.
In both cases,if a trial change is proposed,it will be accepted with the probability
P
acc
m,n
deﬁned earlier.If the change is accepted,LDwill be pursued with the newbonding
scheme (state n) while if the trial move is ﬁnally rejected,LD restarts with the original
bonding scheme corresponding to state m.
31
Figure 2.4:Exemplary conﬁguration of a 7 monomers systemin state n where monomers
3,4 and 5 form a trimer and monomers 6 and 7 a dimer.All pairs of monomers with
mutual distances within the Γ region are indicated by a dotted or a dashed line.In the
text,we consider the Monte Carlo scheme for transitions between states n and m which
only diﬀer by the fact that in state n and mthe 56 pair is respectively open or bounded.
The n →m transition corresponds to the creation of a pentamer by connecting a dimer
and a trimer while the m → n transition leads to the opposite scission.The cross
symbol on link 35 indicates that in state n,when looking to all monomers which could
form a new link with monomer 5,monomer 3 is excluded because it would lead to a
cyclic polymer which is not allowed within the present model.
Coming back to ﬁgure 2.4,we now show that the MC algorithm mentioned above
garantees that the matrix A
trial
mn
is symmetric,an important issue as it leads to the
microreversibility property when combined with the acceptance probabilities described
earlier.Let us deﬁne as P
arm
= 1/2N the probability to select a particular arm,a
uniform quantity.
If conﬁguration m with pair (56) being “bounded” is taken as the starting con
ﬁguration,the number of available arms to form alternative bonds with monomer 5
and monomer 6 are respectively N
5
= 1 and N
6
= 0.Therefore,applying the MC
rules described above,the probability to get conﬁguration n where the pair (56) has to
be unbounded is given by the sum of probabilities to arrive at this situation through
selection of the arm of monomer 5 engaged in the bond with monomer 6 or through
selection of the arm of monomer 6 engaged in a bond with monomer 5.This gives
A
trial
m,n
= P
arm
∗
1
N
5
+1
+P
arm
1
N
6
+1
=
3
2
∗ P
arm
(2.30)
If conﬁguration n with pair (56) being “unbounded” is taken as the starting conﬁg
uration,the application of the MC rules lead to the probability to get conﬁguration m
32
2.3 MonteCarlo procedure
where the pair (56) has to be bounded is given by the sum of probabilities to arrive at
this situation through selection of the free arm of monomer 5 (which has two bounding
possibilities,namely with monomers 2 and 6) or through selection of the free arm of
monomer 6 which can only form a bond with monomer 5.This gives
A
trial
n,m
= P
arm
∗
1
2
+P
arm
=
3
2
∗ P
arm
(2.31)
showing the required matrix symmetry.
33
34
Chapter 3
Equilibrium Properties
We have exploited the model introduced in chapter II in a series of Langevin Dynam
ics (LD) simulations at equilibrium with diﬀerent bonding energy parameter W and
diﬀerent number density φ,and hence,in this chapter,we will present the resulting
equilibrium static and dynamic properties of cylindrical micelles.
Within the static properties,the theoretical prediction of the chain size distribution
has been given by Cates [2] and tested in great detail by Monte Carlo simulation by
Wittmer et al.[13,52] who also investigated conformational properties of chains at
equilibrium.For static properties,our aim is thus mainly to check that our results on
a diﬀerent model are compatible with the analysis of the previous works [13,52].
The two main parameters governing the static properties are the monomer density
φ and the end cap energy E.Our choice of these two parameters is set up with the
aim to simulate three thermodynamic states corresponding to a dilute solution and two
semidilute solutions at the same φ but diﬀerent E,leading to the system with two
average chain lengths ∼ 56 and ∼ 150.
For static properties,the distribution of chain lengths,the gyration radius,and
the endtoend distance versus chain length will be analyzed and compared with pre
vious studies.We have also investigated g
ee
(r) the pair correlation function of end
monomers and P(r) the distribution of bond lengths which are quantitative pertinent
in the microscopic formulation of the kinetic rate constants.
The dynamic properties of the micelles which are given and interpreted in this
chapter form the core of the work.The detailed trajectory of diﬀusing micelles which
are continuously breaking and recombing allows us to analyze the microscopic origin of
rate constants in terms of structural features (e.g.chain end pair correlation function)
35
and dynamic quantities to be related to the statistics of life times of a newly created
chain end.For the latter,we show that the Poisson statistics dominates at long times
while a fraction of correlated recombinations happen at short times.Exploiting these
microscopic features,we characterize the macroscopic scission energy E and the barrier
of recombination B and estimate their values for various state points investigated.
Some macroscopic dynamic properties are then studied with an accent on the mod
iﬁcation of various dynamic relaxation processes due to the scissionrecombination pro
cess.We investigate the monomer diﬀusion and the stress relaxation function.Finally,
we perform a Tjump experiment in order to point out that the previously estimated
macroscopic kinetic constants are indeed the key parameters governing the relaxation
of the chain length distribution.
3.1 Static properties
The main aim of this section is to test the model of chapter II by comparing the struc
tural properties,including the average chain length,the distribution of chain lengths
and the conformations of the chains,with the theory [9,27] and previous simulation
works [13,52].
3.1.1 List of simulation experiments and chain length distribution
The model is studied at three state points.The number of monomers,the number
density φ,the energy parameter W,and the attempt frequency ω per arm are chosen
for
1.A solution at the number density φ = 0.05 and an energy parameter W = 8.
The number of monomers is M=1000.The attempt frequencies of bond scis
sion/recombination per arm ω are 0.1,0.5,1 and 5.This choice will be shown to
lead to a dilute solution.
2.A solution at the number density φ = 0.15 and an energy parameter W = 10.
The number of monomers is M=1000.The attempt frequencies of bond scis
sion/recombination per arm ω are 0.1,0.5,1 and 5.(will be shown to be a
semidilute solution)
36
3.1 Static properties
3.A solution at the number density φ = 0.15 and an energy parameter W = 12.
The number of monomers is M=5000.The attempt frequencies of bond scis
sion/recombination per arm ω are 0.02,0.06,0.1,0.5 and 1 (will be shown to be
a semidilute solution).
Each system evolves according to the Langevin Dynamics algorithm with time step
Δt = 0.005 and is subject to randomtrials of bond scission/recombination with the arm
attempt frequency ω.All the experiments and the results of static properties which
include the average chain length L
0
,the mean square endtoend vector,the radius
of gyration and L
0
/L
∗
,the number of blobs in a chain of length L
0
,are listed in the
table 3.1.Where
R
2
and
R
2
g
are deﬁned as
R
2
L
0
=
L
0
−1
X
n=1
L
0
−1
X
m=1
h¯r
n
¯r
m
i (3.1)
R
2
g
L
0
=
1
2L
2
0
L
0
X
n=1
L
0
X
m=1
(
¯
R
n
−
¯
R
m
)
2
,(3.2)
where ¯r
n
=
¯
R
n+1
−
¯
R
n
.And L
∗
is deﬁned by equation (1.21) (also see section 3.1.2).
As shown in the table,we observed that all the static properties are independent
of ω and therefore,all data can be averaged over all ω values.
3.1.1.1 The dilute case
From Table 3.1,it can be observed that the ﬁrst state point experiment (W = 8,φ =
0.05) is a dilute solution since its average chain length L
0
= 11.48(1) is much smaller
than its crossover value at the monomer number density as calculated by equation (1.20),
L
∗
= 50.5.Dilute solution conditions are conﬁrmed by the chain length distribu
tion shown in ﬁgure 3.1.For dilute conditions,a distribution given by (1.30) is ex
pected [13,52].We ﬁt our data with a single parameter B
1
of function (1.30),where
L
0
is given its computed average value and where γ is given its expected value,γ = 1.165
[13].The ﬁt gives B
1
= 1.08.This curve is signiﬁcantly better than the simple expo
nential distribution expected for ideal or semidilute chain.If γ is left as a second free
parameter in the ﬁt,it gives γ = 1.161 which is also very close to its expected value.
37
Table 3.1:List of simulation experiments and the values of static properties.W is the
scission energy parameter,φ the number density,T
s
the total simulation time,ω the
attempt frequency,L
0
the average chain length,
q
hR
2
i
L
0
the endtoend distance of
the average chain,and
q
R
2
g
L
0
its radius gyration.L
0
/L
∗
is the ratio of the average
chain length over the blob length L
∗
W
φ
T
s
ω
L
0
q
hR
2
i
L
0
q
R
2
g
L
0
L
0
/L
∗
8
0.05
2.5 ∗ 10
5
0.1
11.48(6)
4.93(2)
1.9(2)
0.2
8
0.05
2.5 ∗ 10
5
0.5
11.46(2)
4.92(1)
1.9(1)
0.2
8
0.05
2.5 ∗ 10
5
1.
11.50(3)
4.94(1)
1.9(1)
0.2
8
0.05
2.5 ∗ 10
5
5.
11.51(2)
4.95(1)
1.9(1)
0.2
10
0.15
2.5 ∗ 10
5
0.1
56.2(6)
12.2(1)
4.8(2)
4.7
10
0.15
2.5 ∗ 10
5
0.5
56.2(1)
12.4(1)
4.9(3)
4.7
10
0.15
2.5 ∗ 10
5
1.
56.2(2)
12.3(1)
4.9(3)
4.7
10
0.15
2.5 ∗ 10
5
5.
56.6(2)
12.4(1)
4.9(4)
4.7
12
0.15
6.25 ∗ 10
5
0.02
151(4)
20.9(5)
8.3(4)
12.6
12
0.15
4.5 ∗ 10
5
0.06
153(4)
21.5(7)
8.5(5)
12.6
12
0.15
4.5 ∗ 10
5
0.1
150.7(5)
21.2(2)
8.4(3)
12.6
12
0.15
3 ∗ 10
5
0.5
150.3(5)
20.9(1)
8.4(2)
12.5
12
0.15
3 ∗ 10
5
1
150.4(6)
20.92(3)
8.5(4)
12.5
3.1.1.2 The semidilute case
Both the second state point (W = 10,φ = 0.15) and the third state point (W =
12,φ = 0.15) experiments are found to be in semidilute regime,since their average
chain lengths are 56.4(1) and 151.4(4),respectively,which are several times larger
than their crossover value L
∗
≈ 12 at φ = 0.15 according to equation (1.20).Semi
dilute solution conditions are conﬁrmed by the observation of a simple exponential
distribution of chain lengths.In ﬁgure 3.2 and ﬁgure 3.3,we show,for the second
and the third state point experiments respectively,the agreement of our data with the
expected simple exponential distribution (1.34).
3.1.2 Chain length conformational analysis
In this subsection we are interested in the conformational properties of micelles in
equilibrium and studied as a function of chain size within our polydisperse system.
We have calculated the mean square endtoend distance and the mean square radius
of gyration < R
2
> (L) and < R
2
g
> (L) averaged over subsets of chains of length L.
38
3.1 Static properties
0 1 2 3 4 5 6 7 8 9 10
L/L
0
10
8
10
7
10
6
10
5
10
4
10
3
c0(L)
Figure 3.1:Distribution of chain lengths for the dilute case in good solvent.The data
(squares) are ﬁtted using equation (1.30) with the single parameter B
1
and imposed
value γ = 1.165 (continuous curve).The ﬁt gives B = 1.08.The dashed line shows
the simple exponential distribution c
0
(L) ∝ exp(−L/L
0
) which does not ﬁt the data.
Figure 3.4 and ﬁgure 3.5 are the results for the dilute state point and the two semidilute
state points experiments respectively.As shown in ﬁgure 3.4 relative to the dilute case,
the squares and circles represent our data of < R
2
> and < R
2
g
> respectively.With
the solid line and the dashed line,we indicate,for long chains (L 25),standard
power law scaling L
2ν
with ν = 0.588.The data show a reasonable agreement of this
asymptotic regime in the range of application.
The chains in the semidilute system are expected to behave as ideal chains for
L ≫ L
∗
≈ 12 where L
∗
is estimated from equation.(1.20) with φ = 0.15.Figure 3.5
shows the
R
2
and
R
2
g
of the two semidilute cases.Results of
R
2
and
R
2
g
for the two cases,are superimposed.We indicate for the long chains the ideal chain
conformation
R
2
and
R
2
g
∝ L.The agreement of the ﬁtting lines with simulation
data appears to start at L 60.
The conformational properties of the chain in the two semidilute cases are found
39
0 1 2 3 4 5 6 7
L/L
0
10
8
10
7
10
6
10
5
10
4
c0(L)
Figure 3.2:Distribution of chain lengths for the state point W = 10,φ = 0.15 (ac
cumulated over all ω values).The data (circles) are ﬁtted very well by the simple
exponential function,equation (1.34),with imposed average chain length L
0
= 56.4.
The Dashed line shows the function exp(−1.165L/L
0
) which does not ﬁt the data.
to be identical as expected.The size R of a chain of length L > L
∗
is
R = bL
∗
ν
L
L
∗
0.5
(3.3)
where b is the monomer size and ν is the good solvent scaling exponent while L
∗
(function of φ only) is the same in both cases.
3.1.3 Pair correlation function of chain ends and the distribution of
bond length
Equilibrium polymers are polydisperse polymers endowed with scission and recombi
nation processes.Whereas a scission can happen between any bounded pair,a recom
bination may happen only with two chain ends for linear chains.Thus it is interest
ing to study the spatial distribution of the chain ends and the bond length distribu
tion.For the dilute case the population of bonds ready to open within the Γ range
(−0.96 < r < 1.2) is < N
1Γ
>= 552 and represents 61% of the bonded pairs < N
1
>,
while the population of free arm pairs ready to close,again within the same Γ range,is
40
3.1 Static properties
0 1 2 3 4 5 6
L/L
0
10
8
10
7
10
6
10
5
c0(L)
Figure 3.3:Distribution of chain lengths for the state point W = 12,φ = 0.15 (ac
cumulated over all ω values).The data (circles) are ﬁtted very well by the simple
exponential function,equation 1.34,with imposed average chain length L
0
= 151.4.
The Dashed line shows the function exp(−1.165L/L
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